6  Population Proportions (Intervals Based on Large Samples) (8.4.1 - 8.4.2)

When we use maximum likelihood estimation to find an estimator \(\hat\theta(\boldsymbol{X})\) of some population parameter \(\theta\), we might be interested in constructing a confidence interval around this estimate to give a plausible range of values within which the true value of \(\theta\) lies. In order to do this, we need to make use of the asymptotic properties of maximum likelihood estimators. This states that, provided the sample size \(n\) is sufficiently large, then,

\[ \hat\theta(\boldsymbol{X})\sim N\left(\theta, \sqrt{I_n(\theta)^{-1}}\right) \]

In other words, as \(n\rightarrow\infty\), the expected value of the maximum likelihood estimator is equal to the population parameter it is estimating, \(\theta\), and that the standard deviation of the MLE is equal to \(\sqrt{I_n(\theta)^{-1}}\).

Here, \(I_n(\theta)\) is the Fisher Information for \(\theta\). This can be found as,

\[ I_n(\theta)=-E\left[\frac{\partial^2\ln L(\theta|\boldsymbol{X})}{\partial\theta^2}\right] \]

Knowing the above asymptotic property of maximum likelihood estimators allows us to construct a general \((1-\alpha)\cdot100\%\) asymptotic confidence interval for any MLE as,

\[\begin{equation} \tag{8.37} CI_{1-\alpha}(\theta)=\left[\hat\theta(\boldsymbol{x})-z_{1-\frac{\alpha}{2}}\cdot\sqrt{I_n\big(\hat\theta(\boldsymbol{x})\big)^{-1}},\hat\theta(\boldsymbol{x})+z_{1-\frac{\alpha}{2}}\cdot\sqrt{I_n\big(\hat\theta(\boldsymbol{x})\big)^{-1}}\right] \end{equation}\]

This will allow you to construct a confidence interval for the maximum likelihood estimator of any population parameter, however we are going to focus on confidence intervals for the population proportion of success, denoted by \(\pi\).

6.1 Population Proportions

6.1.1 Asymptotic/Wald Confidence Interval (8.4.1)

If we denote the sample proportion of success by \(p\), then a \((1-\alpha)\cdot100\%\) asymptotic confidence interval for the population proportion \(\pi\), commonly referred to as a Wald interval, is given by,

\[\begin{equation} \tag{8.43} CI_{1-\alpha}(\pi)=\left[p-z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p(1-p)}{n}},p+z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p(1-p)}{n}}\right] \end{equation}\]

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See Section 8.4.1 to see further examples on constructing Wald intervals.

6.1.2 Score/Wilson Confidence Interval (8.4.1.1)

An alternative \((1-\alpha)\cdot100\%\) confidence interval for the population proportion of success \(\pi\) is given by the Wilson interval,

\[\begin{equation} \tag{8.47} \begin{split} CI_{1-\alpha}(\pi)=\Bigg[&\frac{p+\frac{z^2_{1-\alpha/2}}{2n}-z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p(1-p)}{n}+\frac{z^2_{1-\alpha/2}}{4n^2}}}{\left(1+\frac{z^2_{1-\alpha/2}}{n}\right)},\\ &\,\,\,\,\,\,\,\,\frac{p+\frac{z^2_{1-\alpha/2}}{2n}+z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p(1-p)}{n}+\frac{z^2_{1-\alpha/2}}{4n^2}}}{\left(1+\frac{z^2_{1-\alpha/2}}{n}\right)}\Bigg] \end{split} \end{equation}\]

Wilson intervals give a confidence interval with coverage probability which is closer to the \(1-\alpha\) level compared to Wald intervals.

Note that if the sample size is large, then the Wilson interval is roughly equivalent to the Wald interval. This is because with large values of \(n\), \(\frac{z_{1-\alpha/2}^2}{2n}\), \(\frac{z_{1-\alpha/2}^2}{4n^2}\) and \(\frac{z_{1-\alpha/2}^2}{n}\) are all approximately 0.

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See Section 8.4.1.1 of Probability and Statistics with R for more details on this result.

NB: Sections 8.4.1.2, 8.4.1.3 and 8.4.1.4 are not covered/examinable.

6.2 Difference in population proportions (8.4.2)

For random samples of size \(n_X\) and \(n_Y\) taken from two normal distributions, \(X\) and \(Y\) respectively, then the difference in the population proportions of success, \(\pi_X\) and \(\pi_Y\) may be of interest.

A \((1-\alpha)\cdot100\%\) confidence interval for the difference in population proportions, \(\pi_X-\pi_Y\), can be calculated based on the difference in sample proportions, \(p_X-p_Y\), as,

\[\begin{equation} \tag{8.52} \begin{split} CI_{1-\frac{\alpha}{2}}(\pi_X-\pi_Y)=\Bigg[&(p_X-p_Y)-z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p_X(1-p_X)}{n_X}+\frac{p_Y(1-p_Y)}{n_Y}},\\ &\,\,\,\,\,\,\,\,(p_X-p_Y)+z_{1-\frac{\alpha}{2}}\cdot\sqrt{\frac{p_X(1-p_X)}{n_X}+\frac{p_Y(1-p_Y)}{n_Y}}\Bigg] \end{split} \end{equation}\]

If \(|p_X-p_Y|>\frac{1}{2}\left(\frac{1}{n_X}+\frac{1}{n_Y}\right)\), then a continuity correction is generally applied to the above formula. When this condition is satisfied, \(\frac{1}{2}\left(\frac{1}{n_X}+\frac{1}{n_Y}\right)\) is subtracted from the lower limit of the confidence interval above, and added to the upper limit.

The function prop.test() uses this continuity correction automatically whenever the condition is satisfied by the difference in sample proportions.

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See Section 8.4.2 of Probability and Statistics with R for some examples of using this result.